Understanding Sequence Limits: Convergence and Divergence

The video tutorial explains how to determine if a sequence converges or diverges by finding its limit. It covers arithmetic and geometric sequences, using examples to illustrate convergence to a specific number or divergence to infinity. Graphical representations and the squeeze theorem are used to further understand sequence behavior. The tutorial also includes advanced examples involving factorials and natural logarithms.

What is the common difference in the arithmetic sequence defined by a_n = 3n + 4?

• 4

• 3

• 5

• 2

As n approaches infinity, what happens to the sequence a_n = 1/(3^n)?

• It diverges to infinity

• It converges to 0

• It oscillates

• It converges to 1

What is the limit of the sequence a_n = 8n/(3n - 5) as n approaches infinity?

• Infinity

• 0

• 5/3

• 8/3

Which theorem is used to determine the convergence of the sequence sin(n)/n?

• Binomial Theorem

• Squeeze Theorem

• Fundamental Theorem of Calculus

• Pythagorean Theorem

What is the limit of the sequence a_n = ln(n^4)/(5n) as n approaches infinity?

• 4/5

• 1

• Infinity

• 0

How does the sequence n!/(n+1)! behave as n approaches infinity?

• Oscillates

• Diverges to infinity

• Converges to 1

• Converges to 0

What is the limit of the sequence (1 + 1/n)^n as n approaches infinity?

• Infinity

• 0

• e

• 1

What is the result of applying L'Hôpital's rule to the sequence n*sin(1/n)?

• Oscillates

• Diverges to infinity

• Converges to 1

• Converges to 0

What is the limit of the sequence 4n/sqrt(n^2 + 5) as n approaches infinity?

• 2

• 4

• 0

• Infinity

What is the behavior of the sequence n!/(n+2)! as n approaches infinity?

• Diverges to infinity

• Converges to 0

• Oscillates

• Converges to 1